Back K., et al. Stochastic methods in finance (Springer, 2004).pdf

Lecture Notes in Mathematics

1856

Editors:

J.--M. Morel, Cachan

F. Takens, Groningen

B. Teissier, Paris

Subseries:

Fondazione C.I.M.E., Firenze

Adviser: Pietro Zecca

K. Back T.R. Bielecki C. Hipp

S. Peng W. Schachermayer

Stochastic Methods

in Finance

Lecturesgivenatthe

C.I.M.E.-E.M.S. Summer School

held in Bressanone/Brixen, Italy,

July

2003

Editors: M. Frittelli

W. Runggaldier

123

Editors and Authors

Kerry Back

Mays Business School

Department of Finance

310

TX 77879-4218

, USA

Jinan

People’s Republic of China

e-mail: peng@sdu.edu.cn

Wolfgang J. Runggaldier

Dipartimento di Matematica Pura ed Applicata

Universut´adegliStudidiPadova

via Belzoni

e-mail: back@olin.wustl.edu

Tomasz R. Bielecki

Department of Applied Mathematics

Illinois Inst. of Technology

nd Street

, USA

e-mail: bielecki@iit.edu

Marco Frittelli

Dipartimento di Matematica per le Decisioni

Universit ´ adegliStudidiFirenze

via Cesare Lombroso

IL 60616

Padova, Italy

e-mail: runggal@math.unipd.it

Walter Schachermayer

Financial and Actuarial Mathematics

Vienna University of Technology

Wiedner Hauptstrasse

6/17

8/105-1

Firenze, Italy

e-mail: marco.frittelli@dmd.unifi.it

Christian Hipp

Institute for Finance, Banking and Insurance

University of Karlsruhe

Kronenstr.

Vienna, Austria

e-mail: wschach@fam.tuwien.ac.at

Karlsruhe, Germany

e-mail: christian.hipp@wiwi.uni-karlsruhe.de

LibraryofCongressControlNumber:

2004114748

Mathematics Subject Classification (2000):

60G99, 60-06, 91-06, 91B06, 91B16, 91B24, 91B28, 91B30, 91B70, 93-06, 93E11, 93E20

ISSN

3-540-22953-1

Springer-Verlag Berlin Heidelberg New York

DOI:

10.1007

100122

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2004

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Preface

A considerable part of the vast development in Mathematical Finance over

the last two decades was determined by the application of stochastic methods.

These were therefore chosen as the focus of the 2003 School on “Stochastic

Methods in Finance”. The growing interest of the mathematical community in

this ﬁeld was also reﬂected by the extraordinarily high number of applications

for the CIME-EMS School. It was attended by 115 scientists and researchers,

selected from among over 200 applicants. The attendees came from all conti-

nents: 85 were Europeans, among them 35 Italians.

The aim of the School was to provide a broad and accurate knowledge of

some of the most up-to-date and relevant topics in Mathematical Finance.

Particular attention was devoted to the investigation of innovative methods

from stochastic analysis that play a fundamental role in mathematical mod-

eling in ﬁnance or insurance: the theory of stochastic processes, optimal and

stochastic control, stochastic diﬀerential equations, convex analysis and dual-

ity theory.

The outstanding and internationally renowned lecturers have themselves con-

tributed in an essential way to the development of the theory and techniques

that constituted the subjects of the lectures. The ﬁnancial origin and mo-

tivation of the mathematical analysis were presented in a rigorous manner

and this facilitated the understanding of the interface between mathematics

and ﬁnance. Great emphasis was also placed on the importance and eciency

of mathematical instruments for the formalization and resolution of ﬁnancial

problems. Moreover, the direct ﬁnancial origin of the development of some

theories now of remarkable importance in mathematics emerged with clarity.

The selection of the ﬁve topics of the CIME Course was not an easy task be-

cause of the wide spectrum of recent developments in Mathematical Finance.

Although other topics could have been proposed, we are conﬁdent that the

choice made covers some of the areas of greatest current interest.

We now propose a brief guided tour through the topics chosen and through

the methodologies that modern ﬁnancial mathematics has elaborated to unveil

Risk beneath its diﬀerent masks.

Preface

We begin the tour with expected utility maximization in continuous-time

stochastic markets: this classical problem, which can be traced back to the

seminal works by Merton, received a renewed impulse in the middle of the

1980’s, when the so-called duality approach to the problem was ﬁrst devel-

oped. Over the past twenty years, the theory constantly improved, until the

general case of semimartingale stochastic models was ﬁnally tackled with great

success. This prompted us to dedicate one series of lectures to this traditional

as well as very innovative topic:

“Utility Maximization in Incomplete Markets”, Prof. Walter Schachermayer,

Technical University of Vienna.

This course was mainly focused on the maximization of the expected utility

from terminal wealth in incomplete markets. A part of the course was dedi-

cated to the presentation of the stochastic model of the market, with particular

attention to the formulation of the condition of No Arbitrage. Some results of

convex analysis and duality theory were also introduced and explained, as they

are needed for the formulation of the dual problem with respect to the set

of equivalent martingale measures. Then some recent results of this classical

problem were presented in the general context of semi-martingale ﬁnancial

models.

The importance of the above-mentioned analysis of the utility maximization

problem is also revealed in the theory of asset pricing in incomplete markets,

where the agent’s preferences have again to be given serious consideration,

since Risk cannot be completely hedged. Diﬀerent notions of “utility-based”

prices have been introduced in the literature since the middle of the 1990’s.

These concepts determine pricing rules which are often non-linear outside

the set of marketed claims. Depending on the utility function selected, these

pricing kernels share many properties with non-linear valuations: this bordered

on the realm of risk measures and capital requirements. Coherent or convex

risk measures have been studied intensively in the last eight years but only

very recently have risk measures been considered in a dynamic context. The

theory of non-linear expectations is very appropriate for dealing with the

genuinely dynamic aspects of the measures of Risk . This leads to the next

topic:

“Nonlinear expectations, nonlinear evaluations and risk measures”, Prof.

Shige Peng, Shandong University.

In this course the theory of the so-called “ g-expectations” was developed, with

particular attention to the following topics: backward stochastic diﬀerential

equations, F-expectation, g-martingales and theorems of decomposition of E-

supermartingales. Applications to the theory of risk measures in a dynamic

context were suggested, with particular emphasis on the issues of time consis-

tency of the dynamic risk measures.

Among the many forms of Risk considered in ﬁnance, credit risk has received

major attention in recent years. This is due to its theoretical relevance but

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